Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$.  What is the largest of the three numbers?  Express your answer in simplest radical form.
Solution: By Vieta's, $a$, $b$, and $c$ are the solutions to the cubic equation  \[x^3 - 2x^2 - 7x + 14 = 0.\] We group and factor as follows: \begin{align*}
x^3 - 2x^2 - 7x + 14 = 0&=(x^3 - 7x) - (2x^2 - 14)\\
&=x(x^2 - 7) - 2(x^2 - 7)\\
&=(x-2)(x^2 - 7).
\end{align*} Thus, the three solutions are $x=2$, $x=\sqrt{7}$, and $x=-\sqrt{7}$.  The largest of these numbers is $\boxed{\sqrt{7}}$.